True or False. Justify your answer with a proof or a counterexample. Assume all functions $f$ and $g$ are continuous over their domains.

1. If $f(x) < 0,{f}^{\prime }(x) < 0$ for all $x,$ then the right-hand rule underestimates the integral ${\int }_{a}^{b}f(x).$ Use a graph to justify your answer.

Solution

False

2.  ${\int }_{a}^{b}f{(x)}^{2}dx={\int }_{a}^{b}f(x)dx{\int }_{a}^{b}f(x)dx$

3.  If $f(x)\le g(x)$ for all $x\in \left[a,b\right],$ then ${\int }_{a}^{b}f(x)\le {\int }_{a}^{b}g(x).$

Solution

True

4.  All continuous functions have an antiderivative.

Evaluate the Riemann sums ${L}_{4}\text{ and }{R}_{4}$ for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

5.  $y=3{x}^{2}-2x+1$ over $\left[-1,1\right]$

Solution

${L}_{4}=5.25,{R}_{4}=3.25,$ exact answer: 4

6.  $y=\text{ln}({x}^{2}+1)$ over $\left[0,e\right]$

7.  $y={x}^{2} \sin x$ over $\left[0,\pi \right]$

Solution

${L}_{4}=5.364,{R}_{4}=5.364,$ exact answer: 5.870

8.  $y=\sqrt{x}+\frac{1}{x}$ over $\left[1,4\right]$

Evaluate the following integrals.

9.  ${\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx$

Solution

$-\frac{4}{3}$

10.  ${\int }_{0}^{4}\frac{3t}{\sqrt{1+6{t}^{2}}}dt$

11.  ${\int }_{\pi \text{/}3}^{\pi \text{/}2}2 \sec (2\theta ) \tan (2\theta )d\theta$

Solution

1

12.  ${\int }_{0}^{\pi \text{/}4}{e}^{{ \cos }^{2}x} \sin x \cos dx$

Find the antiderivative.

13.  $\int \frac{dx}{{(x+4)}^{3}}$

Solution

$-\frac{1}{2{(x+4)}^{2}}+C$

14.  $\int x\text{ln}({x}^{2})dx$

15.  $\int \frac{4{x}^{2}}{\sqrt{1-{x}^{6}}}dx$

Solution

$\frac{4}{3}\phantom{\rule{0.05em}{0ex}}{ \sin }^{-1}({x}^{3})+C$

16.  $\int \frac{{e}^{2x}}{1+{e}^{4x}}dx$

Find the derivative.

17.  $\frac{d}{dt}{\int }_{0}^{t}\frac{ \sin x}{\sqrt{1+{x}^{2}}}dx$

Solution

$\frac{ \sin t}{\sqrt{1+{t}^{2}}}$

18.  $\frac{d}{dx}{\int }_{1}^{{x}^{3}}\sqrt{4-{t}^{2}}dt$

19.  $\frac{d}{dx}{\int }_{1}^{\text{ln}(x)}(4t+{e}^{t})dt$

Solution

$4\frac{\text{ln}x}{x}+1$

20.  $\frac{d}{dx}{\int }_{0}^{ \cos x}{e}^{{t}^{2}}dt$

The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($) 1980 0 1985 -5,468,750 1990 755,495 1995 -73,005 2000 -29,768 2005 -918 2010 -177 21. If the average cost per gigabyte of RAM in 2010 is$12, find the average cost per gigabyte of RAM in 1980.

Solution

$6,328,113 22. The average cost per gigabyte of RAM can be approximated by the function $C(t)=8,500,000{(0.65)}^{t},$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

23.  Find the average cost of 1GB RAM for 2005 to 2010.

Solution

\$73.36

24.  The velocity of a bullet from a rifle can be approximated by $v(t)=6400{t}^{2}-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?

25.  What is the average velocity of the bullet for the first half-second?

Solution

$\frac{19117}{12}\text{ft/sec},\text{or}1593\text{ft/sec}$